Tennis Elbow
I've gotten tennis elbow in my non-dominant arm when shoveling snow. It goes away if you don't repeat the movement for a long while so it's usually gone by the next time it snows. Tennis players with elbow pain have another problem which is that they want to continue playing tennis. That unfortunately is the one thing they should avoid if it has become severe.
Here are some ideas.
First seek professional medical advice. The ideas below are only augmentative to helping you to keep playing if your doctor agrees to it.
Tendonitis Elbow Brace
Armbands do help cut pain. Look for one that places pressure on the muscle in one place rather than one that covers a whole area. Here is an example.
Switch to a Two handed Backhand
If you are a one handed backhand player there is a good chance you caused the injury hitting backhands so it might be time to start working on a two handed backhand. Technique is often the problem with a one handed backhand elbow injury but once injured even a change in form won't help until you've recovered. A switch to a two hander can often eliminate any shock coming to your dominant arm where you have the injury
Slice Backhand
If you have a one handed backhand and want to stay with it you might try working on a slice backhand since it tends to deliver less shock to the arm.
Flexible Racket
The best way to cut your pain aside from not playing is to find a flexible racket with a stiffness rating below 60 if possible. There are plenty of them out there if you research it. I'd look at one in the 56 range. Here is one of my favorites.
Low Shock Stringing
Reduce your string tension to the minimum and stay away from poly. If you can afford it use gut or at least a gut and synthetic gut hybrid with the mains strung in gut. I'd limit tension to no more than 55 pounds depending of course on the type of frame you use.
Change your Serve
If you have a two handed backhand the chance is that your elbow problems may originate with your serve. Many players cause a problem by using a slice serve with the wrong grip and trying to get excessive wrist snap. Make sure you're using a continental grip or an eastern backhand grip versus an eastern forehand grip when serving with slice. Also, allow your finish to complete on the opposite side of your body so that you aren't putting stress on your arm to stop the follow through.
Try Cold and Heat
I found ice packs and heat pads offer some relief. Often I alternate.
Anti-Inflammatory
Of course this is what the standard suggestion is and the problem is that if you keep playing they won't help much plus they can be tough on your stomach.
Cream
I've found some pain reduction creams do help for joint pain.
My favorite is Aspercreme. You can find it inexpensively at your local department store in the pharmacy section.
The best thing for protecting your arm from further elbow injury is to stop doing the thing that caused the injury. Once you determine that you can change and avoid it and begin to heal. What you never want to do is to get to the point of having to give up the game if possible. It's better to learn a whole new way to play than to lose it due to a repetitive use injury like tennis elbow.
Best of luck in your recovery.